String Theory Demystified

246 String Theory Demystifi ed
The Laws of Black Hole Mechanics
In the early 1970s, James Bardeen, Brandon Carter, and Stephen Hawking found
that there are laws governing black hole mechanics which correspond very closely
to the laws of thermodynamics. 1 The zeroth law states that the surface gravity κ at
the horizon of a stationary black hole is constant.
The fi rst law relates the mass m, horizon area A, angular momentum J, and charge
Q of a black hole as follows:
κ
dm = dA + ΩdJ + Φ dQ
(14.23)
8π
This law is analogous to the law relating energy and entropy. We will see this more
precisely in a moment.
The second law of black hole mechanics tells us that the area of the event horizon
does not decrease with time. This is quantifi ed by writing:
dA ≥ 0 (14.24)
This is directly analogous to the second law of thermodynamics which tells us that
the entropy of a closed system is a nondecreasing function of time. A consequence
of Eq. (14.24) is that if black holes of areas A1 and A coalesce to form a new black
2
hole with area A then the following relationship must hold:
3
A3 > A1+ A2
As you probably recall, an analogous relationship holds for entropy. Finally, we
arrive at the third law of black hole mechanics. This law states that it is impossible
to reduce the surface gravity κ to 0.
The correspondence between the laws of black hole mechanics and
thermodynamics is more than analogy. We can go so far as to say that the analogy
is taken to be real and exact. That is, the area of the horizon A is the entropy S of
the black hole and the surface gravity κ is proportional to the temperature of the
black hole. We can express the entropy of the black hole in terms of mass or area.
In terms of mass the entropy of a black holes is proportional to the mass of the black
1
Bardeen, J.M., B. Carter, and S.W. Hawking, The four laws of black hole mechanics, Comm. Math
Phys. vol. 31, (2), 1973, 161–170.